It is well known, by the Gibbard-Satterthwaite Theorem, that when there are more than two candidates, any non-dictatorial voting rule can be manipulated by untruthful voters. But how strong is the incentive to manipulate under different voting rules? We suggest measuring the potential advantage of a strategic voter by asking how many copies of their (truthful) vote must be added to the election in order to achieve an outcome as good as their best manipulation. Intuitively, this definition quantifies what a voter can gain by manipulating in comparison to what they would have gained by finding like-minded voters to join the election. The higher the former is, the more incentive a voter will have to manipulate, even when it is computationally costly. Using this framework, we obtain a principled method to measure and compare the manipulation potential for different voting rules. We analyze and report this potential for well-known and broad classes of social choice functions. In particular, we show that the positional scoring rule with the smallest manipulation potential will always be either Borda Count (if the number of voters outweighs the number of candidates) or Plurality (vice versa). Further, we prove that any rule satisfying a weak form of majority consistency (and therefore any Condorcet-consistent rule) cannot outperform Plurality, and that any majoritarian Condorcet rule will perform significantly worse. Consequently, out of the voting rules we analyze, Borda Count stands out as the only one with a manipulation potential that does not grow with the number of voters. By establishing a clear separation between different rules in terms of manipulation potential, our work paves the way for the search for rules that provide voters with minimal incentive to manipulate.

翻译：众所周知，根据吉巴德-萨特思韦特定理，当候选人多于两名时，任何非独裁的投票规则都可能被不诚实的选民操纵。但不同投票规则下的操纵动机究竟有多强？我们建议通过以下方式衡量策略型选民的潜在优势：探究需要向选举中添加多少份其（真实）投票的副本，才能获得与其最佳操纵结果同等优质的选举结果。直观而言，该定义量化了选民通过操纵所能获得的收益，相较于通过寻找志同道合的选民加入选举所能获得的收益。前者越高，选民进行操纵的动机就越强，即便计算成本高昂。基于此框架，我们获得了一种原则性方法来度量和比较不同投票规则的操纵潜力。我们针对广泛应用的经典社会选择函数类别进行了分析与报告。特别地，我们证明具有最小操纵潜力的位置计分规则始终为博尔达计数法（当选民数量超过候选人数量时）或多数制（反之）。此外，我们证明了任何满足弱多数一致性条件的规则（因而包括所有孔多塞一致性规则）均无法优于多数制，且任何多数主义孔多塞规则的表现将显著更差。因此，在我们分析的投票规则中，博尔达计数法脱颖而出，成为唯一操纵潜力不随选民数量增长的规则。通过在不同规则的操纵潜力之间建立明确区分，本研究为探索能给予选民最小操纵动机的投票规则开辟了道路。